Gould polynomials

In mathematics, the Gould polynomials ${\displaystyle G_{n}}$, ${\displaystyle n\in \mathbb {N} }$, are polynomials introduced by Henry W. Gould. They are defined in terms of the falling factorial by

${\displaystyle G_{n}(x;a,b)={\frac {x}{x-an}}{\Big (}{\frac {x-an}{b}}{\Big )}_{n}}$.

They form the Sheffer sequence for ${\displaystyle f(t)=e^{at}(e^{bt}-1)}$, ${\displaystyle b\neq 0}$, which has generating function

${\displaystyle \exp(xf^{-1}(t))=\sum _{n=0}^{\infty }G_{n}(x;a,b){\frac {t^{n}}{n!}}.}$

The inverse function is given by

${\displaystyle f^{-1}(t)={\frac {1}{b}}\sum _{k=1}^{\infty }{\binom {-(b+ak)/b}{k-1}}{\frac {t^{k}}{k}}}$.

See also

• Umbral calculus

References

External links

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